# Estimating Bits and Bytes

The following extremely useful relation, specified in IEEE 1541-2002

Powers of 2^10 (1024) are very close to powers of 1000.

        2^0 =                      1 ~ 1000^0
[kibi] 2^10 =                   1024 ~ 1000^1
[mebi] 2^20 =                1048576 ~ 1000^2
[gibi] 2^30 =             1073741824 ~ 1000^3
[tebi] 2^40 =          1099511627776 ~ 1000^4
[pebi] 2^50 =       1125899906842624 ~ 1000^5
[exbi] 2^60 =    1152921504606846976 ~ 1000^6


## Examples

### How many binary search function calls does it take to find one person in the million-person Manhattan phone book

The above lookup table can be used to answer this question.

The binary search is an O(log n) operation, assuming the array is sorted (like a phone book). In this case, we want log base 2 of one million,

${\displaystyle \log _{2}(1,000,000)}$

Log base 10 is easy enough, but we have log base 2. Using the above table, 1M ~ 2^20:

${\displaystyle \log _{2}(1,000,000)\sim log_{2}(2^{20})\sim 20}$

### How many strings with a specified length of 140 characters are in 10 terabytes?

Rumor has it Twitter generates around 10 TB a day. How many 140 character strings would that be, in Python?

Using compact arrays (which Strings are, by default), storing characters as unicode (16 bits, or 2 bytes, per character).

Taking the overly simplistic view that there is around another 100 characters attached with each tweet to indicate username, dates, etc., would make each string:

${\displaystyle 256{\text{unicode characters}}\times {\dfrac {2{\text{bytes}}}{\text{unicode character}}}=512{\text{bytes}}}$

Maybe we're generous and allocate a whole 1000 bytes, to make it 1 KB, per tweet.

Then 10 TB is equivalent to:

${\displaystyle 10Terabytes\times {\dfrac {10^{12}Bytes}{1Terabytes}}\times {\dfrac {1Tweet}{10^{3}Bytes}}\sim 10^{10}}$

That would be 10 billion tweets per day... Several orders of magnitude too high. A useful analysis nonetheless.